Clearly, if \(A\)  has the form as in Equation REF , then it can be constructed by unions and is thus an element of  \(\mathcal {U}_{\infty }\) .
By Lemma REF , \(\mathcal {U}_{\infty }\)  is an algebra and a sigma algebra. It is well known ([1]}, P.2.21) that if \(A\in \mathcal {U}_{\infty }\)  then \(A\)  is a union of some elements in the partition, hence \(A\)  satisfies Equation REF  for some \(\Lambda \)  subset of \(\lbrace 1,2,...,n\rbrace .\) 
To find \(|\mathcal {U}_{\infty }|\) , note that every set \(A \in \mathcal {U}_{\infty }\)  can be written as unions of \(k\)  sets in \(\mathcal {U}\)  and \(k \in \lbrace 0,1,...,n\rbrace \) . The number of ways to select such possible sets is given by \( \binom{n}{0} + \binom{n}{1} + \ldots \binom{n}{n} = 2^{n}.\)
